The two-sheeted hyperboloid $p$ is defined by $x^2 + y^2 - z^2 = -1$
We are given two points, $a = (x_a,y_a,z_a)$ and $b = (x_b,y_b,z_b)$.
$a$, $b$ and $c$ defines the plane $q$.
How can we choose a point $c$ so that the intersection between $p$ and $q$ is a geodesic on the surface of the hyperbola $p$.
The intersection of any plane through the origin with the hyperboloid is a geodesic. All there has to be done, is to make sure that the three points $a$, $b$ and $c$ describe a plane through the origin. If $a$ and $b$ are not parallel, it is easiest to choose $c=(0,0,0)$.